Can anyone help me understand the graph of ellipse and Line

Question

I have an ellipse with semiaxes a and b

Let L denote a line passing through the center of the ellipse and making an angle k with the axis of length 2a.

What does that mean "angle k with the axis of length 2a?"

How can I make a sketch of this graph?

Answer 1

The basic non-rigorous conception of an ellipse is "a stretched out circle". Think of a circle at the center of the xy-plane with a radius of 1. Now I stretch this circle so that on the x-axis, it hits the points (5,0) and (-5,0), and I stretch the height of the circle to (0,3) and (0,-3).

We denote the longer axis as "2a", the major axis, and the shorter axis as "2b" the minor axis. See how in my example the lengths of the axes are 10 and 6? That is because a and b refer to the distance from the center of the ellipse to the edge.

So if there is a line L going through the center of the ellipse you need to label your angle k between the line you just drew and the longer axis...which is the major axis...which has a length of 2a. Let me know if that helped.

*semi-axis just means "half of the axis".

Answer 2

You have a semi-axis of length a, so the complete axis has length 2 x a. The line L is then drawn at an angle k from this axis. Sketching this I would first draw the ellipse. Then from the centre of the ellipse, measure the angle k, and finally sketch L intersecting the centre of the ellipse at that angle.